Proceedings of the 25th Chinese Control Conference

7–11 August, 2006, Harbin, Heilongjiang

Decentralized Steam-Valving Adaptive Neural Control ofMulti-machine Power SystemsXiurong Zhong, Cong Wang, and Yuelong Liu

Control and Optimization Center & College of AutomationSouth China University of Technology, Guangzhou 510641

E-mail: zhongxiurong1981@126.com

Abstract: In this paper, an adaptive neural control scheme is proposed for multi-machine steam-valving power systems.

The multi-machine steam-valving power systems consist of interconnected subsystems, with coupling in the form of

unknown nonlinearities and parametric uncertainties. Using the adaptive neural control method in [11], the coupling

problem between subsystems can be solved without linearization treatment. Semi-global ultimate boundedness of all the

signals in the closed-loop system are obtained. The outputs of the closed-loop system are proven to converge to a small

neighborhood of the desired trajectories. Simulation results of two-machine power systems are presented as an example

to show the effectiveness of the approach.

Key Words: Adaptive neural control, multimachine power systems, steam-valving control, backstepping.

1 INTRODUCTION

Steam-valving control is a very important problem in

power system control. It can not only enhance the sta-

bility of power systems but also increase its dynamic per-

formance. For multi-machine power systems, the steam-

valving control problem is very complicated due to the

couplings among various inputs and outputs, and param-

eter uncertainties. Due to these difficulties, how to design

a steam-valving controller to improve their transient sta-

bility has been become an interest issue for control and

power engineers. Recently, various control technologies

have been applied to steam-valving and excitation con-

trollers of power systems [1]-[4], [5]-[9]. Most of these are

based on differential geometric tools [1]-[3], which can-

cel the inherent system nonlinearities in order to obtain a

feedback equivalent linear system to remove the couplings

of subsystem. With the development of nonlinear theory,

some advanced nonlinear control technologies also have

been applied to excitation and steam-valving controllers of

power systems. In [5][6], decentralized steam valving con-

trol schemes are proposed by using the Hamiltonian func-

tion method. In [7]-[9], various adaptive backstepping con-

trol schemes are proposed to power systems.

In this paper, we present an adaptive neural control scheme

for multi-machine steam-valving power systems using the

approach proposed in [11]. Firstly, we transform the power

systems’ model into a block-triangular form, in which

each machine is considered as one subsystem of the whole

power systems. Secondly, we design a full state feedback

controller for each subsystem utilizing adaptive neural de-

sign. The resulting nonlinear adaptive neural controller

IEEE Catalog Number: 06EX1310

achieves semiglobal uniform ultimate boundedness of all

the signals in the closed-loop system. The outputs of the

system are proven to converge to a small neighborhood of

the desired trajectories. The control performance of the

closed-loop system can be guaranteed by suitably choos-

ing the design parameters.

The rest of the paper is organized as follows: the system

model is introduced in section II. Section III presents the

adaptive neural controller in detail. The uncertainty parts

are approximated by RBF NNS. Simulation results are per-

formed to demonstrate the effectiveness of the designed

controller in section IV. Section V contains the conclusions.

2 MULTIMATION STEAM VALVING SYS-TEM MODEL

Consider an n-machine power system described by the fol-

lowing equation [5]:⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

δi(t) = ωi(t) − ω0

ωi(t) = −Di

Hi[ωi(t) − ω0] + ω0

Hi· [PHi(t)

−GiiE′2qi − Pe − CMiPm0i]

PHi = − 1TH

Pi

PHi(t) + CHi

THP

i

Pmoi + CHi

THP

i

uHi

yi = δi(t)(1)

and

Pe = E′qi ·

n∑j=1

BijE′qjsin(δi(t) − δj(t)) (2)

where

δi : power angle between the q-axis ecletrical potentialvector Eqi and a reference bus voltage vector vREF

in the system in rad;

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ωi : rotating speed of the ith generator, in rad/s;

CMi : power partition isothem of intermediate-pressurecylinder;

PHi : mechanical power of high-pressure(HP) turbine,in per unit;

E′qi : q-axis internal transient electric potential of the

ith generator (assumed as constants in this paper),in per unit;

Pmoi : initial mechanical power of the ith generator, inper unit;

Hi: moment of inertia in second;

Gij : ith row element of nodal susceptance matrix at theinternal nodes after eliminating all phasical buses,in per unit;

Bij : ith row and jth column element of nodal suscep-tance matrix at the internal nodes after eliminatingall phasical buses,in per unit;

THSi: equivalent time constant of HP turbine;

THgi: time constant of oil-servomotor of regulatedvalve of HP turbine;

THi : time constant of HP turbine;

CHi: the power fraction of HP turbine;

uHi : electrical control signal from the controller forthe regulated valve;

Denote ai = Di

Hi, bi = ω0

Hi, ci = ω0

HiCMiPmoi− ω0

HiGijE

′2qi ,

dij = Bijω0Hi

E′qiE

′qj , ei = 1

THP

i

, ki = CHi

THP

i

Pmoi. Let

xi1 = δi(t), xi2 = ωi(t) − ω0, xi3 = PHi(t) as state

variables, ui = CHi

THP

i

uHi as control and THP

i= THSi +

THgi and y(i) as the outputs. Then, Eq. (1) can be rewritten

as follows:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

xi1 = xi2

xi2 = −aixi2 + bixi3 + ci

−∑nj=1 dijsin(xi1 − xj1)

xi3 = −eixi3 + ki + ui

yi = xi1.

(3)

which is in the following block-triangular form

⎧⎪⎪⎪⎨⎪⎪⎪⎩

xi1 = gi1xi2

xi2 = gi2xi3 + fi2(xi2, xj1)xi3 = gi3ui + fi3(xi3)yi = xi1

(4)

where xi2 = [xi1, xi2], xj1(j = 1, 2, · · · , n); fik(k =1, 2) and gim (m = 1, 2, 3) are considered to be unknown

nonlinear smooth functions, and unknown constant param-

eters, respectively.

3 ADAPTIVE NEURAL CONTROL DESIGN

For the adaptive neural controller design of the ith subsys-

tem (3), we employ backstepping method and radial basis

function neural networks (RBFNNs) [10] to design con-

trollers for all the subsystems of (3). An intermediate de-

sired feedback control α∗ij is first shown, then the jth-order

subsystem of the ith subsystem is stabilized with respect

to a Lyapunov function Vij by a stabilizing function αij .

The RBF neural network is used to approximate the un-

known parts in intermediate desired feedback control α∗ij .

The control law ui for the ith subsystem is designed in the

3rd step.

Step 1: Define zi1 = xi1 − xd1. Its derivative is

zi1 = gi1xi2 − xd1 (5)

By viewing xi2 as a virtual control input, there exist a de-

sired feedback control

α∗i1 = −ci1zi1 +

1gi1

xd1 (6)

where ci1 is a positive design constant, gi1 is unknown or

known constant.

Let hi1(Zi1) = −1/gi1xd1 denote the unknown part of

α∗i1, with Zi1 = [xd1]. By employing an RBF neural net-

work WTi1Si1(Zi1) to approximate hi1(Zi1) [12], α∗

i1 can

be expressed as

α∗i1 = −ci1zi1 − W ∗T

i1 Si1(Zi1) − εi1 (7)

where W ∗i1 is the ideal constant weights, and with constant

ε∗i1 > 0, |ε| ≤ ε∗i1 denotes the ideal constant weights.

In practice α∗i1 cannot be realized since W ∗

i1 is unknown.

Therefore, define zi2 = xi2 − αi1 and

αi1 = −ci1zi1 − WTi1Si1(Zi1) (8)

Then, we have

zi1 = gi1(zi2 + αi1) − xd1

= gi1[zi2 − ci1zi1 − WTi1Si1(Zi1) + εi1] (9)

Consider the Lyapunov function candidate

Vi1 =1

2gi1z2i1 +

12WT

i1Γ−1i1 Wi1, (10)

whose derivative is

Vi1 =zi1zi1

gi1+ WT

i1Γ−1i1 Wi1

=zi1zi2 − ci1z2i1 + zi1εi1

− WTi1Si1(Zi1)zi1 + WT

i1Γ−1i1

˙Wi1

(11)

Consider the adaption law for Wi1 as

˙Wi1 = ˙Wi1 = ΓWi1 [Si1(Zi1)zi1 − σi1Wi1] (12)

1163

where σi1 > 0 and ΓWi1 = ΓTWi1

> 0 are design constants,

Wi1 = Wi1 − W ∗i1. By completion of suqares, we have

−σi1WTi1Wi1 = −σi1W

Ti1(Wi1 + W ∗

i1)

≤ −σi1‖Wi1‖2

2+

σi1‖W ∗i1‖2

2(13)

−ci1z2i1 + zi1εi1 ≤ ε∗2i1

4ci1.

Then we have the following inequality

Vi1 < zi1zi2 − σi1‖Wi1‖2

2+

σi1‖W ∗i1‖2

2+

ε∗2i1

4ci1. (14)

Step 2: Define zi2 = xi2 − αi1. Its derivative is

zi2 = gi2xi3 + fi2(xi2, xj1) − αi1 (15)

By viewing xi3 as a virtual control to stabilize the (zi1, zi2)subsystem of the ith subsystem, there exists a desired feed-

back control

α∗i2 = −zi1 − ci2zi2 − 1

gi2(fi2 − αi1) (16)

where ci2 > 0 is a design constant, αi1 is a function of

xd1, xi1 and Wi1. Therefore, αi1 can be described as

αi1 =∂αi1

∂xi1(gi1xi2) + Φi1

where

Φi1 =∂αi1

∂xd1xd1 +

∂αi1

∂Wi1

[Γi1(Si1(Zi1)zi1 − σi1Wi1)]

is computable. Let hi2(Zi2) = 1/gi2(fi2− αi1) denote the

unknown part of α∗i2 with Zi2 = [xi1, xj1, xi2, Φi1], i �=

j. By utilizing an RBF neural network to approximate

hi2(Zi2), then

α∗i2 = −zi1 − ci2zi2 − WT

i2Si2(Zi2) − εi2

(17)

where W ∗i2 denotes the ideal constant weights, and |εi2| ≤

ε∗i2 is the approximation error with constant ε∗i2 > 0. How-

ever, W ∗i2 is unknown and α∗

i2 cannot be realized in prac-

tice. Let us define zi3 = xi3 − αi2 and

αi2 = −zi1 − ci2zi2 − WTi2Si2(Zi2). (18)

Then, we have

zi2 = gi2[zi3 − zi1 − ci2zi2 − WTi2Si2(Zi2) + εi2] (19)

Consider the Lyapunov function candidate

Vi2 = Vi1 +1

2gi2z2i2 +

12WT

i2Γ−1i2 Wi2. (20)

whose derivative is

Vi2 = Vi1 +zi2zi2

gi2+ WT

i2Γ−1i2

˙Wi2. (21)

Consider the adaption law for Wi2 as

˙Wi2 = ˙Wi2 = ΓWi2 [Si2(Zi2)zi2 − σi2Wi2] (22)

where σi2 > 0 and ΓWi2 = ΓTWi2

> 0 are design constants,

Wi2 = Wi2 − W ∗i2. Then Vi2 becomes

Vi2 = Vi1 − zi1zi2 + zi2zi3 − ci2z2i2

+ zi2εi2 − σi2WTi2Wi2. (23)

By completion of squares, we have

−σi2WTi2Wi2 = −σi2W

Ti2(Wi2 + W ∗

i2)

≤ −σi2‖Wi2‖2

2+

σi2‖W ∗i2‖2

2

−ci2z2i2 + zi2εi2 ≤ ε∗2i2

4ci2. (24)

Then, we have the following inequality:

Vi2 < zi2zi3 −2∑

k=1

σi2‖Wik‖2

2+

2∑k=1

σi2‖W ∗ik‖2

2+

2∑k=1

ε∗2ik

4cik.

(25)

Step 3: The derivative of zi3 = xi3 − αi2 is

zi3 = gi3ui + fi3(xi3) − αi2 (26)

To stabilize the ith subsystem (zi1, zi2, zi3), there exists a

desired feedback control

u∗i = −zi2 − ci3zi3 − 1

gi3(fi3 − αi2) (27)

where ci3 > 0 is a design constant to be specified later. αi2

is a function of xi1, xi2, xj1, xd1, Wi1 and Wi2. So αi2 can

be expressed as

αi2 =∂αi2

∂xi1xi1 +

∂αi2

∂xi2xi2 +

∂αi2

∂xj1xj1 + Φi2 (28)

where

Φi2 =∂αi2

∂xd1xd1 +

2∑k=1

∂αi2

∂Wik

˙Wik (29)

For the desired control u∗i , let the unknown part hi3(Zi3) =

1/gi3(fi3 − αi2) of u∗i is approximated by an RBF neural

network WTi3Si3(Zi3) and εi3 ≤ ε∗i3 is the approximation

error with constant ε∗i3 > 0.

Because W ∗i3 is unknown, u∗

i cannot be realized in practice.

Consider

ui = −zi2 − ci3zi3 − WTi3Si3(Zi3). (30)

Then, we have

zi3 = gi3[−zi2 − ci3zi3 − WTi3Si3(Zi3) + εi3].

Consider the Lyapunov function candidate

Vi3 = Vi2 +1

2gi3z2i3 +

12WT

i3Γ−1i3 Wi3 (31)

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The derivative of Vi3 is

Vi3 = Vi2 − zi2zi3 − ci3z2i3 + zi3εi3

− WTi3Si3(Zi3)zi3 + WT

i3Γ−1i3

˙Wi3. (32)

Consider the adaption law for Wi3 as

˙Wi3 = ˙Wi3 = ΓWi3 [Si3(Zi3)zi3 − σi3Wi3] (33)

where σi3 > 0 and ΓTWi3

= ΓWi3 are de-

sign constants, Wi3 = Wi3 − W ∗i3, Zi3 =

[xi2, xi3, xj2, xi1, xj1, Φi2]T , i �= j. Then

Vi3 = Vi2 − zi2zi3 − ci3z2i3 + zi3εi3 − σi3W

Ti3Wi3.

(34)

By completion of squares, we have

−σi3WTi3Wi3 ≤ −σi3‖Wi3‖2 + σi3‖Wi3‖‖W ∗

i3‖

≤ −δi3‖Wi3‖2

2+

σi3‖W ∗i3‖2

2

−ci3z2i3 + zi3εi3 ≤ ε2i3

4ci3. (35)

If we choose cik such that cik ≥ γi/2gik, where γi is a

positive constant, and choose σik and Γik such that Γik ≥γiλmaxΓ−1

ik , k = 1, 2, 3. Then, we have the following in-

equality:

Vi3 < −3∑

k=1

cikz2ik −

3∑k=1

σik‖W‖2

2+ δi

≤ −ri[3∑

k=1

12gik

z2ik +

3∑k=1

WTikΓ−1

ik Wik

2] + δi

≤ −riVi3 + δi (36)

where

δi ≤3∑

k=1

σik‖W ∗ik‖2

2+

3∑k=1

ε∗2ik

4cik(37)

Let V =∑n

i=1 Vi3. Its derivative is

V =n∑

i=1

Vi3 <n∑

i=1

(−riVi3 + δi) < −rV + δ (38)

where r = min{r1, . . . , rn} and δ =∑n

i=1 δi are posi-

tive constants. According to lemma 1.2 in [11], all zik and

Wik(i = 1, . . . , n, k = 1, 2, 3) are uniformlly bounded

for bounded initial conditions. Since xd1, zi1, xi1 and

Wi1 are bounded, we have that αi1 is also bounded. From

zi2 = xi2 − αi1, xi2 is bounded. Using the same method,

we can have αi2, xi3, xj1 and ui are bounded. Therefore,

all the signals in the closed-loop system remain bounded.

4 SIMULATION STUDIES

In this section, we consider two-machine (n = 2) power

systems. Select the reference model yd1 = xd1 =0.35(sin(3t) + 1). The control objective is to design neu-

ral controllers for this coupling power system such that the

outputs yi1(i = 1, 2) follow the desired trajectory yd1.

According to Section III, the controllers for the two sub-

systems are chosen as:

u1 = −z12 − c13z13 − WT13S13(Z13)

u2 = −z22 − c23z23 − WT23S23(Z23) (39)

where

zi1 = xi1 − yd1 = xi1 − xd1

zi2 = xi2 − αi1

zi3 = xi3 − αi2, (i = 1, 2)

with

αi1 = −ci1zi1 + WTi1Si1(Zi1)

Z11 = Z21 = [xd1]

Φi1 =∂αi1

∂xd1xd1 +

∂αi1

∂Wi1

[Γi1(Si1(Zi1)zi1 − σi1Wi1)]

Zi2 = [xi1, xj1, xi2, Φi1] ∈ R4

αi2 = −zi1 − ci2zi2 − WTi2Si2(Zi2).

Φi2 =∂αi2

∂xd1xd1 +

2∑k=1

∂αi2

∂Wik

˙Wik

Zi3 = [xi2, xi3, xj2, xi1, xj1, Φi2]T ∈ R6

˙Wik = ΓWik

[Sik(Zik)zik − σikWik](i = 1, 2; j = 1, 2; k = 1, 2, 3, i �= j)

The parameters of the two-machine power system are cho-

sen as follows:

parameters generator 1 generator 2

xd 1.863 2.36

x′d 0.257 0.319

xT 0.129 0.11

D 5 3

T′do 6.9 7.96

H/s 8 10.2

xad/s 1.712 1.712

kc 1 1

w0/(rad.s−1) 314.159 314.159

CM 0.7 0.72

CH 0.3 0.29

Pm0 0.82 0.8

THP

i0.398 0.4

In our simulation, neural network WTi1Si1(Zi1)(i = 1, 2)

contains 21 nodes with centers Ui1(i = 1, 2) evenly

spaced in [0, 0.7], and widths ηi1 = 2. Neural network

WTi2Si2(Zi2)(i = 1, 2) contains 81 nodes with centers

1165

Figure 1: output δ11 follow the reference xd1(rad)

Figure 2: output δ21 follow the reference xd1(rad)

evenly spaced in [−1, 1] × [−1, 1] × [−1, 1] × [−1, 1] and

width ηi2 = 2(i = 1, 2). Neural network WTi3Si3(Zi3)(i =

1, 2) contains 729 nodes with centers evenly spaced in

[−1, 1]× [−1, 1]× [−1, 1]× [−1, 1]× [−1, 1]× [−1, 1] and

width ηi3 = 2(i = 1, 2). Increasing parameters cik(i =1, 2; k = 1, 2, 3) will increase the stability time. The ini-

tial condition are X110 = 0.27, X210 = 0.25, X120 =0.55, X220 = 0.55, X130 = 0.53, X230 = 0.53, Xd1 =0.35, Wi10 = Wi20 = Wi30(i = 1, 2) and σi1 = σi2 =σi3 = 0.05(i = 1, 2), cik = 10.5(i = 1, 2; k = 1, 2, 3).

The simulation results are shown in Figure 1 and Figure 2.

5 CONCLUSION

In this paper, an adaptive neural control scheme has been

proposed for multi-machine stream-volving power sys-

tems. With the help of neural networks to approximated all

the unknown part of the power systems, the designed con-

trollers can control effectively the multi-machine stream-

volving power systems.

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